= Boltzmann carrier density of a two-dimensional parabolic band
{c}
{title2=$n=gmk_BT\,e^{-(E_c-\mu)/(k_BT)}/(2\pi\hbar^2)$}
State counting with total degeneracy $g$ gives a constant two-dimensional <density of states> $gm/(2\pi\hbar^2)$ above a parabolic <conduction band> edge. Integrating the Boltzmann occupation yields the displayed areal <carrier density>. For one valley and two spin states the coefficient is $mk_BT/(\pi\hbar^2)$. Applying the same count to <holes> gives $np=N_cN_v e^{-E_g/(k_BT)}$ and the intrinsic density $n_i=\sqrt{N_cN_v}e^{-E_g/(2k_BT)}\propto T e^{-E_g/(2k_BT)}$, distinct from the three-dimensional $T^{3/2}$ prefactor.
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